Instability of Reissner–Nordström black hole in Einstein-Maxwell-scalar theory
Abstract
The scalarization of Reissner–Nordström black holes was recently proposed in the Einstein-Maxwell-scalar theory. Here, we show that the appearance of the scalarized Reissner–Nordström black hole is closely related to the Gregory-Laflamme instability of the Reissner–Nordström black hole without scalar hair.
1 Introduction
Recently, the scalarized black hole solutions were found from Einstein-scalar-Gauss-Bonnet (ESGB) theories [1, 2]. We note that these black holes with scalar hair are connected to the appearance of instability for the Schwarzschild black hole without scalar hair. Interestingly, the instability of Schwarzschild black hole in ESGB theory is regarded as not the tachyonic instability but the Gregory-Laflamme (GL) instability [3] by comparing it with the instability of the Schwarzschild black hole in the Einstein-Weyl gravity [4].
The notion of the GL instability comes from the three observations [5, 6, 7, 8]: (1) the instability is based on the \(s(l=0)\)-mode perturbations for scalar and tensor fields. (2) The linearized equation includes an effective mass term, providing that the potential develops negative region near the black hole horizon but it becomes positive after crossing the r-axis. (3) The instability of a black hole without hair is closely related to the appearance of a newly black hole with hair.
More recently, a scalarization of the Reissner–Nordström (RN) black hole was proposed in the Einstein-Maxwell-scalar (EMS) theory which is considered as a simpler theory than the ESGB theory [9]. We note that the scalarized black holes were found in the Einstein-scalar-Born-Infeld theory [10, 11], regarded as a generalized EMS theory. The EMS theory includes three physically propagating modes of scalar, vector, and tensor. In this case, the instability of RN black hole is determined solely by the linearized scalar equation because the RN black hole is stable against tensor-vector perturbations, as found in the Einstein-Maxwell theory [12, 13, 14, 15].
In this work, we wish to show that the appearance of the scalarized RN black hole is closely associated with the GL instability of the RN black hole without scalar hair. Here, the GL instability will be determined by solving the linearized scalar equation. This will indicate an important connection between scalarized RN black holes and GL instability of RN black holes.
The organization of our work is as follows. We introduce the EMS theory and its linearized theory around the RN black hole background in Sect. 2. In Sect. 3, we perform the stability analysis for the RN black hole based on the linearized scalar Eq. (18). Mainly, we derive the GL instability bound (20). We solve the static linearized Eq. (22) to confirm the threshold of the instability \(\alpha _{\mathrm{th}}\) as well as to obtain \(n=0,1,2\cdots \) scalarized RN black holes in Sect. 4. In Sect. 5, we explore what the GL instability is. Section 6 is devoted to obtaining a scalarized RN black hole by solving the four Eqs. (41)–(44) numerically. It indicates that the appearance of the scalarized RN black hole is closely related to the GL instability of the RN black hole without scalar hair. Also, we obtain scalarized RN black holes for the quadratic coupling of \(\alpha \phi ^2 \) for comparison to the exponential coupling \( e^{\alpha \phi ^2}\). Finally, we will describe our main results in Sect. 7.
2 EMS and its linearized theory
3 Instability of RN black hole
In analyzing the stability of the RN black hole in the EMS theory, we first consider the two linearized Eqs. (7) and (8) because two perturbations of metric \(h_{\mu \nu }\) and vector \(a_{\mu }\) are coupled. Exactly, these correspond to the linearized equations for the Einstein-Maxwell theory. For the odd-parity perturbations, one found the Zerilli-Moncrief equation which describes two physical DOF propagating around the RN background [12, 13]. Also, the even-parity perturbations with two physical degrees of freedom (DOF) were studies in [14, 15]. It turns out that the RN black hole is stable against these perturbations. In this case, a massless spin-2 mode starts with \(l=2\), while a massless spin-1 mode begins with \(l=1\). The EMS theory provides \(5 (=2+2+1)\) DOF propagating around the RN background.
At this stage, we would like to mention that such potentials exist around neutral black holes (black holes without charge) in higher dimensions and the S-deformation has been used to confirm the stability of neutral black holes in higher dimensions [20]. We conjecture that the GL instability may occur for \(\alpha _\mathrm{th}> 19.83\), but the threshold of instability \(\alpha _\mathrm{th}\) should be determined explicitly by the numerical computations. We expect that \(\alpha _\mathrm{th}\) is located at the shaded region between \(\alpha _\mathrm{in}\) and \(\alpha _\mathrm{po}\). Usually, if the potential V derived from physically propagating modes is negative in some region, a growing perturbation can appear in the spectrum. This might indicate an instability of the black hole system under such perturbations. However, this is not always true. Some potentials with negative region near the horizon do not imply the instability. The criterion to determine whether a black hole is stable or not against the perturbation is whether the time-evolution of the perturbation is decaying or not. The perturbed equation around a RN black hole can usually be described by the Schrödinger-type equation (18), where a growing mode like \(e^{\Omega t}\) of the perturbation indicates the instability of the black hole. The absence of any unstable physical fields provides a precise way of determining the stability of the black hole.
4 Static scalar perturbation
5 GL instability
The instability of the RN black hole may be regarded as the GL instability since this instability is based on the \(s(l=0)\) mode of a perturbed scalar and its linearized equation includes an effective mass term (not tachyonic mass of \(m^2_\mathrm{t}<0\) presicely) which develops negative potential near the horizon from the Maxwell kinetic term. In this section, we wish to clarify the similarity and difference between the GL instability (modal instability) and tachyonic instability because the instability of RN black hole is closely related to appearance of scalarized RN black holes.
Gregory-Laflamme (GL) instability among RN black hole (RNBH) in EMS theory and and Schwarzschild black hole (SBH) in Einstein-Weyl gravity. LR denotes Licherowicz-Ricci tensor
Theory | Einstein-Maxwell-scalar theory | Einstein-Weyl gravity |
---|---|---|
Action | \(S_\mathrm{EMS}\) in (1) | \(S_\mathrm{EW}\) in (30) |
BH without hair | RNBH with \(\bar{\phi }=0\) | SBH with \(\bar{R}_{\mu \nu }=0\) |
Linearized equation | Scalar equation (9) | LR-equation (31) |
GL instability mode | s-mode of \(\varphi \) | s-mode of \(\delta R_{\mu \nu }\) |
Bifurcation points | \(\alpha =8.019,40.84,99.89,\cdots \) for \(q=0.7\) | \(m_2^2=0.7677\) |
Potential and its asymptotic form | V(r) in (15) and \(V_{r\rightarrow \infty }=0\) | \(V_Z(r)\) in (35) and \(V_{Z,r\rightarrow \infty }=m_2^2\) |
GL instability bound | \(\alpha >\alpha _\mathrm{th}(q)\) in (20) | \(0<m_2<\frac{0.876}{r_+}\) in (36) |
Small unstable BH | \(r_+<r_c=1.714\) with \(\alpha _\mathrm{th}=8.019(q=0.7)\) | \(r_+<r_c=0.876\) with \(m_\mathrm{th}=1\) |
BH with hair | Scalarized RN BH | Non-Schwarzschild BH |
6 Scalarized RN black holes
6.1 Exponential coupling
Before we proceed, we note that the RN black hole solution is allowed for any value of \(\alpha \), while a scalarized RN black hole solution may exist only for \(\alpha \ge \alpha _\mathrm{th}\). The threshold of instability for a RN black hole reflects the disappearance of zero crossings in the perturbed scalar profiles. We explore a close connection between the instability of a RN black hole without scalar hair and appearance of a scalarized RN black hole. As a concrete example, we wish to find a scalarized RN black hole which is closely related to the \(q=0.7(M=1,~Q=0.7)\) and \(\alpha \ge 8.019\) case (\(n=0\) case).
For the RN black hole with \(\phi _0=0\), the outer horizon is located at \(r_{+}=1.714\) and the charge-mass ratio is given by \(q=0.7\). In the Fig. 8 (left), one observes that for given \(\alpha =8.019\), the ratio of q for the \(n=0\) scalarized RN black hole increases beyond the extremal RN black hole (\(q=1\)) as \(\phi _0\) increases. Moreover, in the Fig. 8 (right), the scalar at the horizon \(\phi _0\) increases as the horizon radius \(r_+\) decreases. The scalar at the horizon is terminated at \(r_+=1.714\), corresponding to the RN outer horizon. It is the starting point for a scalarized RN black hole, while from (21) it corresponds to the ending point for unstable RN black hole.
6.2 Quadratic coupling
Considering the quadratic coupling of \(\alpha \phi ^2\), we have to choose \(\bar{\phi }=\mathrm{const}\) to obtain the RN black hole with different charge \(\tilde{Q}^2=\alpha \bar{\phi }^2 Q^2\). In order to make the analysis simple, we may choose an equivalent coupling of \(1+\alpha \phi ^2\) with \(\bar{\phi }=0\) to give the RN black hole. In this case, the bifurcation points of the RN solution are the same as those of exponential coupling because the static scalar equation takes the same form as in (22). Furthermore, instabilities of RN solution are exactly the same for both couplings. To obtain scalarized RN black holes, we solve Eqs. (41)–(44) by replacing \(e^{\alpha \phi ^2}\) with \(1+\alpha \phi ^2\). From Figs. 8, 9, 10, we observe that the quadratic coupling shows the similar properties to the exponential coupling.
As was mention in [26], however, the only difference between two coupling in the ESGB theory is that the \(n=0\) fundamental branch of scalarized black holes is stable for the exponential coupling, while the \(n=0\) fundamental branch is unstable for the quadratic coupling. Therefore, we expect that the similar thing will happen since the \(n=0\) scalarized RN black hole turned out to be unstable in the EMS theory with exponential coupling [27].
7 Discussions
First of all, we mention that scalarized RN black holes were found in the EMS theory. It is emphasized that the appearance of these black holes with scalar hair is closely related to the instability of the RN black hole without scalar hair in the EMS theory. Concerning the appearance scalarized RN black holes [9], it is very important to obtain the precise threshold \(\alpha _\mathrm{th}\) of instability for the RN black hole in the EMS theory. In this work, we have obtained the GL instability bound (20) for the RN black hole in the EMS theory by considering \(s(l=0)\)-mode scalar perturbation.
Roughly speaking, a shape of scalar potential V(r) in (15) determines the instability of RN black hole. The sufficient condition of \(\int ^{\infty }_{r_+} dr [V(r)/f(r)]<0\) for instability [18, 19] gives rises to an analytic bound (16), while the sufficient condition for stability is given by the other bound (17). Explicitly, for \(q=0.418\), the sufficient condition for instability takes the form of \(\alpha >30.74\), whereas the sufficient condition for the stability is given by \(0<\alpha \le 19.83\). In the case of \(\int ^{\infty }_{r_+}dr [V(r)/f(r)]>0\) with negative potential near the horizon, however, it is not easy to make a clear decision on the stability of the black hole. Here it is still stable for \(19.83<\alpha \le 29.47\) with \(q=0.418\) even providing negative region near the horizon shown in Fig. 1. In this case, the S-deformation method might provide a complementary result to support the stability of such black holes by finding the deformed potential [21, 22].
In general, the GL instability bound is not given by an analytic form. As was shown in Fig. 3 depending on q, it was determined by solving the linearized Eq. (9) numerically. In the case of \(q=0.418\), the GL instability bound is \(\alpha > 29.47\) which is surely less than the sufficient condition for instability (\(\alpha >30.74\)). Importantly, this picture shows that the GL instability appeared in a simpler EMS theory than the ESGB theory and Einstein-Weyl gravity. For \(q=0.7\), we have obtained the GL instability bound of \(\alpha >\alpha _\mathrm{th}= 8.019\). We have derived the precise value of threshold \(\alpha _\mathrm{th}=8.019\) again by solving the static linearized equation numerically. Furthermore, we have obtained the \(n=0(\alpha \ge 8.019)\) scalarized RN black hole by solving Eqs. (41)–(44) numerically for exponential and quadratic couplings.
Consequently, we have explored a clear connection between GL instability of RN black hole and scalarization of RN black hole.
Notes
Acknowledgements
This work was supported by the National Research Foundation of Korea (NRF) Grant funded by the Korea government (MOE) (No. NRF-2017R1A2B4002057).
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